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On the index of length four minimal zero-sum sequences. (English) Zbl 1301.11016

Summary: Let \(G\) be a finite cyclic group. Every sequence \(S\) over \(G\) can be written in the form \(S=(n_1g)\cdot\ldots\cdot(n_lg)\) where \(g\in G\) and \(n_1, \ldots, n_l\in[1, \text{ord}(g)]\), and the index \(\text{ind}(S)\) is defined to be the minimum of \((n_1+\cdots+n_l)/\text{ ord}(g)\) over all possible \(g\in G\) such that \(\langle g \rangle =G\). A conjecture says that every minimal zero-sum sequence of length 4 over a finite cyclic group \(G\) with \(\gcd(|G|, 6)=1\) has index 1. This conjecture was confirmed recently for the case when \(|G|\) is a product of at most two prime powers. However, the general case is still open. In this paper, we make some progress towards solving the general case. We show that if \(G=\langle g\rangle\) is a finite cyclic group of order \(|G|= n\) such that \(\gcd(n,6)=1\) and \(S=(x_1g)\cdot(x_2g)\cdot(x_3g)\cdot(x_4g)\) is a minimal zero-sum sequence over \(G\) such that \(x_1,\dots,x_4\in[1,n-1]\) with \(\gcd(n,x_1,x_2,x_3,x_4)=1\), and \(\gcd(n,x_i)>1\) for some \(i\in[1,4]\), then \(\text{ind}(S)=1\). By using a new method, we give a much shorter proof to the index conjecture for the case when \(|G|\) is a product of two prime powers.

MSC:

11B50 Sequences (mod \(m\))
20K01 Finite abelian groups
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